def test_sin():
v1 = ad.create_vector('v', [0, 100])
v2 = ad.sin(v1)
assert(v2[0].getValue() == 0)
assert(np.isclose(v2[1].getValue(), np.sin(100)))
jacobian = ad.get_jacobian(v2, ['v1', 'v2'])
assert(np.array_equal(jacobian, np.array([[1, 0], [0, np.cos(100)]])))
v1 = ad.Scalar('x', 4)
v2 = ad.Scalar('y', 10)
v3 = ad.sin(np.array([v1, v2])) / ad.sin(np.array([v1, v2]))
assert(np.isclose(v3[0].getValue(), 1))
assert(np.isclose(v3[1].getValue(), 1))
jacobian = ad.get_jacobian(v3, ['x', 'y'])
assert(np.isclose(jacobian, np.array([[0, 0], [0, 0]])).all())
v1 = ad.Scalar('x', 4)
v2 = ad.Scalar('y', 10)
v3 = ad.sin(np.array([v1, v2])) ** 2
assert(np.isclose(v3[0].getValue(), np.sin(4) ** 2))
assert(np.isclose(v3[1].getValue(), np.sin(10) ** 2))
jacobian = ad.get_jacobian(v3, ['x', 'y'])
assert(np.isclose(jacobian, np.array([[2 * np.sin(4) * np.cos(4), 0], [0, 2 * np.sin(10) * np.cos(10)]])).all())
v1 = ad.Scalar('x', 4)
v2 = ad.Scalar('y', 10)
v3 = ad.sin(np.array([v1 * v2, v1 + v2])) ** 2
assert(np.isclose(v3[0].getValue(), np.sin(40) ** 2))
assert(np.isclose(v3[1].getValue(), np.sin(14) ** 2))
jacobian = ad.get_jacobian(v3, ['x', 'y'])
assert(np.isclose(jacobian, np.array([[2 * np.sin(40) * np.cos(40) * 10, 2 * np.sin(40) * np.cos(40) * 4],
[2 * np.sin(14) * np.cos(14), 2 * np.sin(14) * np.cos(14)]])).all())
def run_problem4():
print("-" * 18 + " Problem 4 " + "-" * 18)
x1 = ad.Variable()
x2 = ad.Variable()
x3 = ad.Variable()
y = ((ad.sin(x1 + 1) + ad.cos(2 * x2)) * ad.tan(ad.log(x3)) +
(ad.sin(x2 + 1) + ad.cos(2 * x1)) * ad.exp(1 + ad.sin(x3)))
x1_f = np.random.rand()
x2_f = np.random.rand()
x3_f = np.random.rand()
x1_v = noise_like(x1_f)
x2_v = noise_like(x2_f)
x3_v = noise_like(x3_f)
f, grad = ad.func(y, {x1: x1_f, x2: x2_f, x3: x3_f}, get_gradient=True)
f_np = ((np.sin(x1_f + 1) + np.cos(2 * x2_f)) * np.tan(np.log(x3_f)) +
(np.sin(x2_f + 1) + np.cos(2 * x1_f)) * np.exp(1 + np.sin(x3_f)))
print("Function value by autodiff =", f)
print("Function value by numpy =", f_np)
lhs = (ad.func(y, {
x1: x1_f + t * x1_v,
x2: x2_f + t * x2_v,
x3: x3_f + t * x3_v
}) - f) / t
rhs = (np.sum(grad[x1] * x1_v) + np.sum(grad[x2] * x2_v) +
np.sum(grad[x3] * x3_v))
print("(V(w + tv)-V(w)) / t =", lhs)
print("<dV(w), v> =", rhs)
print("|lfs - rhs| / |lhs| =", np.abs(lhs - rhs) / np.abs(lhs))
def test_cos():
x = ad.Scalar('x', 2)
val = ad.cos(x)
assert (np.isclose(val.getValue(), -0.41614683654))
assert (np.isclose(val.getDeriv()['x'], -0.9092974268256817))
x = ad.Scalar('x', 2)
val = ad.cos(2 * x)
assert (np.isclose(val.getValue(), np.cos(4)))
assert (np.isclose(val.getDeriv()['x'], -2 * np.sin(4)))
x = ad.Scalar('x', -2)
val = ad.cos(x)
assert (np.isclose(val.getValue(), -0.4161468365471424))
assert (np.isclose(val.getDeriv()['x'], 0.9092974268256817))
x = ad.Scalar('x', 0)
val = ad.cos(x)
assert (np.isclose(val.getValue(), 1.0))
assert (np.isclose(val.getDeriv()['x'], 0.0))
x = ad.cos(ad.Scalar('x', 100))
x2 = ad.sin(ad.Scalar('x', 100))
val = x**2 + x2**2
assert (np.isclose(val.getValue(), 1))
assert (np.isclose(val.getDeriv()['x'], 0.0))
assert (ad.cos(0) == 1)
assert (ad.cos(13443) == np.cos(13443))
def test_sin():
from autodiff import sin
x = AdFloat(np.pi)
f = lambda x: sin(x) * 2
assert f(x).dx == np.cos(np.pi) * 2
assert f(x).x == np.sin(np.pi) * 2
def test_nonlinfit():
from pyfitting import nonlinfit
from autodiff import sin
f = lambda x, p: p[0]*sin(x)
x = np.arange(0,100,1.0/1000)
fit = nonlinfit(f, x, f(x,[10]),[10])
np.testing.assert_almost_equal(fit['p'][0], 10)
def test_sin():
x = ad.Scalar('x', 2)
val = ad.sin(x)
assert (np.isclose(val.getValue(), 0.90929742682))
assert (np.isclose(val.getDeriv()['x'], -0.41614683654))
x = ad.Scalar('x', 2)
val = ad.sin(2 * x)
assert (np.isclose(val.getValue(), np.sin(4)))
assert (np.isclose(val.getDeriv()['x'], 2 * np.cos(4)))
x = ad.Scalar('x', 2)
val = 3 * ad.sin(x) + 10 * x + 5
assert (np.isclose(val.getValue(), 3 * np.sin(2) + 25))
assert (np.isclose(val.getDeriv()['x'], 3 * np.cos(2) + 10))
x = ad.Scalar('x', 2)
y = ad.Scalar('y', 5)
val = 3 * ad.sin(x) + 10 * ad.sin(y)
assert (np.isclose(val.getValue(), 3 * np.sin(2) + 10 * np.sin(5)))
assert (np.isclose(val.getDeriv()['x'], 3 * np.cos(2)))
assert (np.isclose(val.getDeriv()['y'], 10 * np.cos(5)))
x = ad.Scalar('x', -2)
val = ad.sin(x)
assert (np.isclose(val.getValue(), -0.9092974268256817))
assert (np.isclose(val.getDeriv()['x'], -0.4161468365471424))
x = ad.Scalar('x', 0)
val = ad.sin(x)
assert (np.isclose(val.getValue(), 0.0))
assert (np.isclose(val.getDeriv()['x'], 1.0))
assert (ad.sin(0) == 0)
assert (ad.sin(13443) == np.sin(13443))
def test_composite():
#Test some more complicated functions / identities, including some multivariate ones.
x = ad.Scalar('x', 2)
z = (5 * (x + 20) / 10)**2
d = z.getGradient(['x'])
assert (z.getValue() == 121)
assert (np.array_equal(d, [11]))
x = ad.Scalar('x', 2)
y = ad.Scalar('y', 3)
z = (x + 20) * y
d = z.getGradient(['x', 'y'])
assert (z.getValue() == 66)
assert (np.array_equal(d, [3, 22]))
x = ad.Scalar('x', 1)
y = ad.Scalar('y', 3)
z = (x * y + x) * y
d = z.getGradient(['x', 'y'])
assert (z.getValue() == 12)
assert (np.array_equal(d, [12, 7]))
x = ad.Scalar('x', 2)
y = ad.Scalar('y', 3)
z = (x + y) / y
d = z.getGradient(['x', 'y'])
assert (np.isclose(z.getValue(), 5 / 3))
assert (np.allclose(d, [1.0 / 3, -2.0 / 9]))
x = ad.Scalar('x', 2)
y = ad.Scalar('y', 3)
z = x + ((y**2) / y)
d = z.getGradient(['x', 'y'])
assert (z.getValue() == 5)
assert (np.array_equal(d, [1, 1]))
x = ad.Scalar('x', 2)
y = ad.Scalar('y', 3)
z = (x**y)**2
d = z.getGradient(['x', 'y'])
assert (z.getValue() == 64)
assert (np.array_equal(d, [6 * 32, 2 * np.log(2) * 64]))
x = ad.Scalar('x', 16)
y = ad.sin(ad.sqrt(x))
assert (np.isclose(y.getValue(), np.sin(4)))
assert (np.isclose(y.getDeriv()['x'], 1 / 8 * np.cos(4)))
#trig identity
x = ad.Scalar('x', 5)
y = ad.sin(x)**2 + ad.cos(x)**2
assert (np.isclose(y.getValue(), 1))
assert (np.isclose(y.getDeriv()['x'], 0))
#trig identity multivariate
x = ad.Scalar('x', 5)
y = ad.Scalar('y', 243423)
z = ad.sin(x * 1231 * y)**2 + ad.cos(x * 1231 * y)**2
assert (np.isclose(z.getValue(), 1))
assert (np.isclose(z.getDeriv()['x'], 0))
assert (np.isclose(z.getDeriv()['y'], 0))
x = ad.Scalar('x', 16)
y = ad.sqrt(ad.power(x, 2))
assert (y.getValue() == 16)
assert (y.getDeriv()['x'] == 1)
x = ad.Scalar('x', 10)
y = ad.tan(x) * ad.cos(x) / ad.sin(x)
assert (np.isclose(y.getValue(), 1))
assert (np.isclose(y.getDeriv()['x'], 0))
#https://math.berkeley.edu/~kruckman/fall2010/9-29-problems.pdf
x = ad.Scalar('x', 10)
y = (x**2 * ad.sin(x) / (x**2 + 1))
assert (np.isclose(y.getValue(), 100 * np.sin(10) / (101)))
assert (np.isclose(y.getDeriv()['x'],
(20 * np.sin(10) +
(10**4 + 100) * np.cos(10)) / (101**2)))
x = ad.Scalar('x', 4)
y = (x**3 * ad.exp(x))
assert (np.isclose(y.getValue(), 4**3 * (np.exp(4))))
assert (np.isclose(y.getDeriv()['x'],
3 * 16 * np.exp(4) + 4**3 * np.exp(4)))
x = ad.Scalar('x', 4)
y = ad.sin(x) * ad.cos(x) * ad.tan(x)
assert (np.isclose(y.getValue(), np.sin(4) * np.cos(4) * np.tan(4)))
assert (np.isclose(y.getDeriv()['x'], np.sin(8)))
x = ad.Scalar('x', 4)
y = ad.sqrt(x) / ad.tan(x)
assert (np.isclose(y.getValue(), 2 / np.tan(4)))
assert (np.isclose(y.getDeriv()['x'],
1 / (np.tan(4) * 4) - 2 / (np.sin(4)**2)))
x = ad.Scalar('x', 4)
y = ad.exp(ad.sqrt(x + 1))
assert (np.isclose(y.getValue(), np.exp(5**0.5)))
assert (np.isclose(y.getDeriv()['x'], np.exp(5**0.5) / (2 * (5**0.5))))
x = ad.Scalar('x', 4)
y = ad.exp(ad.sin(ad.exp(x)))
assert (np.isclose(y.getValue(), np.exp(np.sin(np.exp(4)))))
assert (np.isclose(y.getDeriv()['x'],
np.cos(np.exp(4)) * np.exp(np.sin(np.exp(4)) + 4)))
x = ad.Scalar('x', 4)
y = (ad.sin(x**(1 / 3)))**(1 / 3)
assert (np.isclose(y.getValue(), np.sin(4**(1 / 3))**(1 / 3)))
assert (np.isclose(
y.getDeriv()['x'],
np.cos(4**(1 / 3)) / (9 * ((4 * np.sin(4**(1 / 3)))**(2 / 3)))))
x = ad.Scalar('x', 16)
y = ad.Scalar('y', 9)
z = ad.sqrt(x * y)
assert (z.getValue() == 12)
assert (np.isclose(z.getDeriv()['x'], 9 / 2 * (16 * 9)**(-0.5)))
assert (np.isclose(z.getDeriv()['y'], 16 / 2 * (16 * 9)**(-0.5)))
x = ad.Scalar('x', 16)
y = ad.Scalar('y', 9)
z = ad.sqrt(x * (y**4))
assert (z.getValue() == 4 * 81)
assert (np.isclose(z.getDeriv()['x'], (9**4) / 2 * (16 * (9**4))**(-0.5)))
assert (np.isclose(z.getDeriv()['y'],
(4 * 16 * (9**3)) / 2 * (16 * (9**4))**(-0.5)))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
z = ad.cos(ad.sin(x * y))
assert (np.isclose(z.getValue(), np.cos(np.sin(6))))
assert (np.isclose(z.getDeriv()['x'], -2 * np.cos(6) * np.sin(np.sin(6))))
assert (np.isclose(z.getDeriv()['y'], -3 * np.cos(6) * np.sin(np.sin(6))))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
x = ad.log(ad.Scalar('x', 10), 10)
x2 = ad.log(ad.Scalar('y', 100), 10)
x3 = x - x2
x4 = ad.log(ad.Scalar('x', 10) / ad.Scalar('y', 100), 10)
assert (np.isclose(x3.getValue(), -1))
assert (np.isclose(x3.getDeriv()['x'], 1 / (np.log(10) * 10)))
assert (np.isclose(x3.getDeriv()['y'], -1 / (np.log(10) * 100)))
assert (np.isclose(x3.getValue(), x4.getValue()))
assert (np.isclose(x3.getDeriv()['x'], x4.getDeriv()['x']))
assert (np.isclose(x3.getDeriv()['y'], x4.getDeriv()['y']))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
z = ad.sin(2 * x + ad.exp(y**2) + 4 * x * y)
assert (np.isclose(z.getValue(), np.sin(6 + np.exp(4) + 24)))
assert (np.isclose(z.getDeriv()['x'], np.cos(6 + np.exp(4) + 24) * (10)))
assert (np.isclose(z.getDeriv()['y'],
np.cos(6 + np.exp(4) + 24) * (4 * np.exp(4) + 12)))
#http://math.gmu.edu/~memelian/teaching/Fall08/partDerivExamples.pdf
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
z = x * ad.exp(2 * x + 3 * y)
assert (np.isclose(z.getValue(), 3 * np.exp(12)))
assert (np.isclose(z.getDeriv()['x'], 6 * np.exp(12) + np.exp(12)))
assert (np.isclose(z.getDeriv()['y'], 9 * np.exp(12)))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
z = (x - y) / (x + y)
assert (np.isclose(z.getValue(), 1 / 5))
assert (np.isclose(z.getDeriv()['x'], 4 / 25))
assert (np.isclose(z.getDeriv()['y'], -6 / 25))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
z = ad.Scalar('z', 5)
f = x * ad.cos(z) + ((x**2) * (y**3) * ad.exp(z))
assert (np.isclose(f.getValue(), 3 * np.cos(5) + (9 * 8 * np.exp(5))))
assert (np.isclose(f.getDeriv()['x'], np.cos(5) + 2 * 3 * 8 * np.exp(5)))
assert (np.isclose(f.getDeriv()['y'], 3 * 9 * 4 * np.exp(5)))
assert (np.isclose(f.getDeriv()['z'], -3 * np.sin(5) + 9 * 8 * np.exp(5)))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
f = (y / x) * ad.ln(x)
assert (np.isclose(f.getValue(), 2 / 3 * np.log(3)))
assert (np.isclose(f.getDeriv()['x'], 2 / 9 * (1 - np.log(3))))
assert (np.isclose(f.getDeriv()['y'], 1 / 3 * np.log(3)))
x = ad.Scalar('x', 3)
y = ad.Scalar('y', 2)
f = 1 / (x**2 + y**2)
assert (np.isclose(f.getValue(), 1 / 13))
assert (np.isclose(f.getDeriv()['x'], -6 / (13**2)))
assert (np.isclose(f.getDeriv()['y'], -4 / (13**2)))
def f(x, p):
return p[0] * (x**2) + p[1] * x + p[2] * sin(x)